Packing of regular tetrahedral quartets of circles on a sphere

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 459, Issue 2039, 2003, Pages 2847-2859

  Tarnai, T ^a   Fowler, P W ^b   Kabai, S ^c  

^a BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

^b UNIVERSITY OF EXETER   (United Kingdom)

^c Uniconstant Company   (Hungary)

Author keywords

Compound of tetrahedra; Constrained Tammes problem; Optimization; Rigidity of packing; Spherical circle packing; Symmetry

Indexed keywords

ALGORITHM; ARTICLE; CALCULATION; COMPUTER; GEOMETRY; RIGIDITY; THEORY;

EID: 1542407749     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2003.1160     Document Type: Article

References (41)

1. Aste, A. & Weaire, D. 2000 The pursuit of perfect packing. Bristol: Institute of Physics Publishing.
2. Baldwin, J. 1997 Buckyworks: Buckminster Fuller's ideas for today. Wiley.
3. Brückner, M. 1900 Vielecke und Vielflache. Leipzig: Teubner.
4. Conway, J. H. & Sloane, N. J. A. 1999 Sphere packings, lattices and groups, 3rd edn. Springer.
5. Conway, J. H., Hardin, R. H. & Sloane, N. J. A. 1996 Packing lines, planes, etc.: packings in Grassmannian spaces. Exp. Math. 5, 139-159.
6. Coxeter, H. S. M. 1973 Regular polytopes. New York: Dover.
7. Cromwell, P. 1997 Polyhedra. Cambridge University Press.
8. Cundy, H. M. & Rollett, A. P. 1981 Mathematical models, 3rd edn. Stradbroke: Tarquin.
9. 2 with minimum distance as large as possible. Discrete Math. 60, 3-66.)
10. 2 with minimum distance as large as possible. Discrete Math. 60, 3-66.)
11. Fejes Tóth, L. 1943 Über die Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems. Jahresber. Dtsch. Math.-Verein. 53, 66-68.
12. Fejes Tóth, L. 1953 Lagerungen in der Ebene auf der Kugel und im Raum. Springer.
13. Fejes Tóth, L. 1964 Regular figures. New York: Macmillan.
14. Fowler, P. W. & Tarnai, T. 1996 Transition from spherical circle packing to covering: geometrical analogues of chemical isometrization. Proc. R. Soc. Lond. A 452, 2043-2064.
15. Fowler, P. W., Cremona, J. E. & Steer, J. I. 1988 Systematics of bonding in non-icosahedral carbon clusters. Theor. Chim. Acta, 73, 1-26.
16. Fowler, P. W., Tarnai, T. & Gáspár, Zs. 2002 From circle packing to covering on a sphere with antipodal constraints. Proc. R. Soc. Lond. A 458, 2275-2287.
17. Gillespie, R. J. & Nyholm, R. S. 1957 Inorganic stereochemistry. Q. Rev. Chem. Soc. 11, 339-380.
18. Goldberg, M. 1937 A class of multi-symmetric polyhedra. Tôhoku Math. J. 43, 104-108.
19. Greenwood, N. N. & Earnshaw, A. 1986 Chemistry of the elements. Oxford: Pergamon.
20. Grünbaum, B. 1967 Convex polytopes, p. 235. Wiley.
21. Hargittai, I. & Hargittai, M. 1986 Symmetry through the eyes of a chemist. Weinheim: VCH.
22. Holden, A. 1991 Shapes, space and symmetry. New York: Dover.
23. Lee, S. H. 1994 Arrangements of 13 and 14 points on the unit sphere with maximum minimum distance. PhD thesis, University of California at Berkeley, California, USA.
24. Leksell, L. 1983 Stereotactic radiosurgery. J. Neurol. Neurosurg. Psychiatry 46, 797-803.
25. Linnett, J. W. 1964 The electronic structure of molecules. London: Methuen.
26. Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical recipes. The art of scientific computing. Cambridge University Press.
27. Robinson, R. M. 1961 Arrangement of 24 points on a sphere. Math. Annln 144, 17-48.
28. Robinson, R. M. 1969 Finite sets of points on a sphere with each nearest to five others. Math. Annln 179, 296-318.
29. Sadoc, J. F. & Mosseri, R. 1997 Frustration géomé trique. Paris: Éditions Eyrolles.
30. Schütte, K. & van der Waerden, B. L. 1951 Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand ein Platz? Math. Annln 123, 96-124.
31. Sloane, N. J. A., Hardin, R. H., Smith, W. D. et al. 2000 Tables of spherical codes. Published electronically at http://www.research.att.com/~njas/ packings/.
32. Tammes, P. M. L. 1930 On the origin of number and arrangement of the places of exit on the surface of pollen-grains. Recl Trav. Bot. Néerlandais 27, 1-84.
33. Tarnai, T. 1984 Spherical circle-packing in nature, practice and theory. Struct. Topol. 9, 39-58.
34. Tarnai, T. 2002 Polymorphism in multi-symmetric close packings of equal spheres on a spherical surface. Struct. Chem. 13, 289-295.
35. Tarnai, T. & Gáspár, Zs. 1983 Improved packing of equal circles on a sphere and rigidity of its graph. Math. Proc. Camb. Phil. Soc. 93, 191-218.
36. Tarnai, T. & Gáspár, Zs. 1987 Multi-symmetric close packings of equal spheres on the spherical surface. Acta Crystallogr. A 43, 612-616.
37. Tarnai, T. & Gáspár, Zs. 1991 Arrangement of 23 points on a sphere (on a conjecture of R. M. Robinson). Proc. R. Soc. Lond. A 433, 257-267.
38. Verheyen, H. F. 1996 Symmetry orbits. Boston: Birkhäuser.
39. Wenninger, M. J. 1971 Polyhedron models. Cambridge University Press.
40. Wenninger, M. J. 1983 Dual models. Cambridge University Press.
41. Wilson, J. 1993 The memorial by Nicholas Stone to Sir Thomas Bodley. Church Monum. 8, 57-62.